Back in the day I had a little bag specifically for all basic nail care needs; I still do the same now as it's easy to grab but since then a lot of the products have changed...
For base coat needs I'm currently favouring Rimmel's Nail Nurse Stronger Nail - I've had no staining from darker colours with this and colour 'sticks' well to it. Coupled with my favoured top coat, Seche Vite - they are a forced to be reckoned with.
I adore the Sally Hansen Instant Cuticle Remover (but I do prefer the maximum strength version,) for ridding of pesky cuticles - for which I just use a standard cuticle stick. No fancy tools needed for that job.
Sally Hansen also does the best nail scissors, from what I've found - nice and sturdy as I can't abide nail clippers. I also keep a spare of tweezers (for rogue hang nails or splinters,) and also a crystal nail file - they are so much better for the nails than standard emery boards! Buffers are also pretty crucial to get a nice smooth base and I'm quite enamored with this one.
Lastly, cuticle oil. Mine tend to get pretty dry so this is a crucial step. I also like to keep a few separators around for pedicures - lord knows how much of a pain it is to paint toes without them...
So, am I missing anything?